We establish a Galois correspondence for finite quantum groupoid actions on
II_1 factors and show that every finite index and finite depth subfactor is an
intermediate subalgebra of a quantum groupoid crossed product. Moreover, any
such a subfactor is completely and canonically determined by a quantum groupoid
and its coideal *-subalgebra. This allows to express the bimodule category of a
subfactor in terms of the representation category of a corresponding quantum
groupoid and the principal graph as the Bratteli diagram of an inclusion of
certain C^*-algebras related to it.